COLUMBIA, South Carolina (CNN) -- Gov. Mark Sanford, who has admitted having an extramarital affair with an Argentine woman, met Friday with members of his Cabinet, saying he wanted to apologize to them personally.

Gov. Mark Sanford, who admitted having an extramarital affair, apologized to his Cabinet.

"I had not yet had a chance to apologize to you all personally. I wanted to do so," the Republican governor said.

"I've made general apologies to the people of South Carolina or to staff at large, but you all are really the epicenter of that team that ... through about 65,000 state employees serves the people of South Carolina. And so I wanted generally to apologize to every one of you all for letting you down."

Disappearing from the public eye for nearly a week, Sanford, 49, acknowledged Wednesday that he had not hiked the Appalachian Trail as his staff had said earlier, but had been in Buenos Aires, Argentina.

Sanford said Thursday that he would reimburse South Carolina for the Argentina leg of a state-funded trade mission last year because he saw the woman he had an affair with on that trip. He described the woman as "a dear, dear friend."

CNN's sources in Buenos Aires and in South Carolina identified the woman, whose name was also widely reported in Argentine media, as Maria Belen Chapur.

In his apology to the Cabinet, the governor referenced the biblical figure David, noting that he "fell mightily, fell in very significant ways, but then picked up the pieces."

He also addressed members of the staff by name, saying his behavior had put them in awkward positions.

Sanford's most recent amends came as a state representative called for a grand jury investigation into the "possible misuse of state property, finances, abuses of power and the negligence of duties in the Office of Governor."

State Rep. Boyd Brown, a Democrat, wrote state Attorney General Henry McMaster that the questions that have arisen from the governor's statements and from media reports present "troubling" issues.

A key South Carolina Republican cited Sanford's past criticism of Bill Clinton and accused the governor of hypocrisy.

"He was saying our elected leaders need to stand firm on principles and values, and one of those is strong family values," Glenn McCall, a member of the Republican National Committee, told CNN by phone.

"What he said is hypocritical if he doesn't step down, because he was right with what he said about Clinton and others. When you are an elected leader, we hold you to higher standards."

Meanwhile, a second South Carolina newspaper Friday called on Sanford to resign.

The editorial Friday morning in the Orangeburg Times and Democrat comes two days after Sanford admitted to the extramarital affair. The governor also acknowledged Wednesday that he did not tell his staff that he was in Argentina during a five-day period over the past week.

"Sanford should exit not because of his personal problems but because in dealing with those problems, he has shown clearly that he cannot be an effective leader. The entire episode vividly illustrates irresponsibility and poor judgment. He could not have handled the situation worse," the paper says in an editorial.

"Sanford decided Wednesday to step aside as leader of the Republican Governors Association. If he cannot be an effective leader of that organization, how can he expect to be effective as the state's chief executive? Sanford indicated Wednesday he will remain in office. He should not," the editorial said.

On Thursday, the Spartanburg Herald-Journal became the first newspaper in South Carolina to call on Sanford to resign.

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"And the state needs a leader it can trust as it deals with the troubled economy. Sanford has destroyed that trust," it said.

A key South Carolina Republican cited Sanford's past criticism of Bill Clinton and accused the governor of hypocrisy.

"He was saying our elected leaders need to stand firm on principles and values, and one of those is strong family values," Glenn McCall, a member of the Republican National Committee, told CNN by phone.

The State, the Columbia-based newspaper that acquired what it said were e-mail exchanges between Sanford and the woman, acknowledged Thursday that there would likely be people who would call for the governor's resignation.

"We are not ready to join them at this point," an editorial in the newspaper said.

The State said it acquired the e-mails in December. The governor's office confirmed their authenticity Wednesday, the newspaper told CNN. When contacted by CNN, a spokesman for the governor would neither confirm nor deny the authenticity of the e-mails.

The affair began in the past year and was discovered five months ago, Sanford said Wednesday without elaborating. He added that he and his wife were trying to work through it. Watch CNN.com bloggers discuss situation »

He implied that he had ended the affair, saying, "And the one thing that you really find is that you absolutely want resolution. And so oddly enough, I spent the last five days of my life crying in Argentina." Watch iReporters sound off »

Thursday, the governor said he would repay South Carolina for the Argentina leg of the trade mission last year with the Department of Commerce.

"While the purpose of this trip was an entirely professional and appropriate business development trip, I made a mistake while I was there in meeting with the woman who I was unfaithful to my wife with," Sanford said.

"That has raised some very legitimate concerns and questions, and as such I am going to reimburse the state for the full cost of the Argentina leg of this trip." Watch report from CNN's Jessica Yellin »

The South American swing took Sanford and several commerce officials to Brazil and Argentina for one week, beginning on June 21, 2008.

According to state expenditure reports, Sanford's expenses for out-of-state travel with the Department of Commerce were $21,487 for the fiscal year ending June 30, 2008. Governors commonly travel out-of-state or abroad to stir up investment in their home states.

It was not immediately clear how much of the expenses the Argentina part of the trip comprised.

Sanford said in the news conference Wednesday that he footed the bill for his most recent trip to Buenos Aires, which occurred in the past week.